Breakdown of the Bretherton law due to wall slippage

Against the common wisdom that wall slip plays only a minor role in global flow characteristics, here we demonstrate theoretically for the displacement of a long bubble in a slippery channel that the well-known Bretherton 2/3 law can break down due to a fraction of wall slip with the slip length lambda much smaller than the channel depth R. This breakdown occurs when the film thickness h(infinity) is smaller than lambda, corresponding to the capillary number Ca below the critical value Ca* similar to (lambda/R)(3/2). In this strong slip regime, a new quadratic law h(infinity)/R similar to Ca-2(R/lambda)(2) is derived for a film much thinner than that predicted by the Bretherton law. Moreover, both the 2/3 and the quadratic laws can be unified into the effective 2/3 law, with the viscosity mu replaced by an apparent viscosity mu(app) = mu h(infinity)/(lambda + h(infinity)). A similar extension can also be made for coating over textured surfaces where apparent slip lengths are large. Further insights can be gained by making a connection with drop spreading. We find that the new quadratic law can lead to theta(d) proportional to Ca-1/2 for the apparent dynamic contact angle of a spreading droplet, subsequently making the spreading radius grow with time as r proportional to t(1/8). In addition, the precursor film is found to possess l(f) proportional to Ca-1/2 in length and therefore spreads as l(f) proportional to t(1/3) in an anomalous diffusion manner. All these features are accompanied by no-slip-to-slip transitions sensitive to the amount of slip, markedly different from those on no-slip surfaces. Our findings not only provide plausible accounts for some apparent departures from no-slip predictions seen in experiments, but also offer feasible alternatives for assessing wall slip effects experimentally.